Pasha Zusmanovich - teaching

A.T. Fomenko, Double cover of the Klein bottle by torus

Pasha Zusmanovich

6/7TOPO Topology, winter semester 2025/2026

Exam term:

Monday February 2, 2026 09:00 G503

Rules for taking exams

LITERATURE
Main:

J.R. Munkres, Topology, 2nd ed., Prentice Hall, 2000
O.Ya. Viro, O.A. Ivanov, N.Yu. Netsvetaev, V.M. Kharlamov, Elementary Topology. Problem Textbook, AMS, 2008 Referred in what follows as [V]

Additional:

A.V. Arkhangelskii, V.I. Ponomarev, Fundamentals of General Topology: Problems and Exercises, D. Reidel, 1984
R. Engelking, General Topology, revised and completed ed., Heldermann Verlag, 1989
A. Fomenko, D. Fuchs, Homotopical Topology, 2nd ed., Springer, 2016 (included here solely for its pictures)
M.W. Hirsch, Differential Topology, Springer, 2001
J.F. Kelley, General Topology, Van Nostrand, 1955; reprinted by Springer, 1975 and Dover, 2017
C. Kosniowski, A First Course in Algebraic Topology, Cambridge Univ. Press, 1980 (Russian edition only)
J.W. Milnor, Topology from the Differentiable Viewpoint, The University Press of Virginia, 1965
S.A. Morris, Topology Without Tears
M. Starbird, F. Su, Topology Through Inquiry, MAA Press, 2019

All the books are available in electronic form in multiple places.

= available in the university library

A BRIEF SYNOPSIS
(Each class lasted approximately 1.5 hours, unless specified otherwise; tutorials are running separately)

Class 1 September 22, 2025
Organizational details. The subject of topology. Definition of a topological space. Examples of topological spaces: trivial and discrete toplogy, topologies on 2- and 3-element sets, the standard topology on \(\mathbb R\). Number of topologies on a finite set.
([V], pp. xi-xii, 2, 11-12; Munkres, p.76).

Class 2 September 29, 2025
Base of a topology, examples. Any base of the standard topology on \(\mathbb R\) can be decreased. Criterion for a family of open sets to be a base ([V], p.16, 3.A).

Class 3 October 6, 2025
Further criteria for a family of subsets to be a base ([V], 3.B, 3.C). Finer and coarser topologies. Criterion for a topology to be coarser (Munkres, Lemma 13.3).
([V], pp.16-17; Munkres, p.81).

Class 4 October 13, 2025
The standard topology on \(\mathbb R^2\). Closed sets, definition of topology in terms of closed sets. Clopen sets. A parody of an episode from the movie "The Bunker" featuring clopen sets.
([V], pp.13-14,16-17).

Class 5 October 20, 2025
Descritption of topologies having exactly one base. Metric spaces. Euclidean metric, Manhattan metric, 0-1 metric, finite metric spaces. Restriction of a metric to a subspace. Balls and spheres. Metric topology.
([V], pp.16,18-20,22; Munkres, pp.119-120).

Class 6 October 27, 2025
Proof that the metric topology is well-defined (open balls form a base of the topology). Examples of metric topologies. Metrizability of topological spaces, examples of metrizable and non-metrizable spaces. Hausdorff topological space. Metric topologies are Hausdorff. The arrow.
([V], pp. 11-12,22,97; Munkres, pp.119-123).

Class 7 November 10, 2025 (3 hours)
Topological and metric equivalences of metrics. Metric equivalence implies topological equivalence. If \(\rho\) is a metric, then \(\frac{\rho}{1+\rho}\) is topologically equivalent, but not necessary metrically equivalent, metric.

Subspace topology, examples. Interior, closure, boundary.
([V], pp.22-23,27,29-31).

Class 8 November 24, 2025 (3 hours)
The boundary of a set coincides with the set of boundary points (remake). Definition of continuous and open (i.e., an image of an open set is open) maps. \(x \mapsto |x|\) is continuous but not open. The continuity of a map \(f: X \to Y\) is equivalent to each of the following conditions: \[ f^{-1}(Cl(B)) \supseteq Cl(f^{-1}(B)) \text{ for any } B \subseteq Y \\ f^{-1}(Int(B)) \subseteq Int(f^{-1}(B)) \text{ for any } B \subseteq Y \\ f(Cl(A)) \subseteq Cl(f(A)) \text{ for any } A \subseteq X \\ f(Int(A)) \supseteq Int(f(A)) \text{ for any } A \subseteq X . \] Continuity at a point. Equivalence with the \(\epsilon-\delta\) definiton of continuity from analysis in the case of metric topologies. Homeomorphisms (definition, elementary properties).
([V], pp.30,59-61,67-69; Munkres, pp.102-105).

Class 9 December 8, 2025
Further examples and non-examples of homeomorphisms. (Non)homeomorphisms between planes with punctures. Conectedness, connected component.
([V], pp.69-73,83-85).

Class 10 December 15, 2025
Connected subsets of \(\mathbb R\) with the standard topology are exactly all kinds of intervals, and singletons. The image of a continuous map of a connected topological space is connected. Topological proof of the Intermediate Value Theorem from real analysis. The product topology. The quotient topology.
([V], pp.84,87,89,136-137,141-143,145-148; Munkres, pp.86-87).

EXAM
Set 1:

Set 2:

Set 3:

Set 4:

Set 5:

Set 6:

Set 7:

A brief synopsis, videos, and homeworks from this course at the previous semesters

Created: Fri Oct 2 2020
Last modified: Mon Dec 22 2025 14:44:28 CET